To ensure that the contact lines are pinned on the plate, we set the value of \(C\)  on the hot bottom plate equal to the initial value as the boundary condition of the phase field. The other boundary conditions are no-slip velocities on the top and bottom plates, fixed temperature \(\theta _{cold}=0\)  (top) and \(\theta _{hot}=1\)  (bottom), and periodic conditions in the horizontal directions. Stretched grids with \(288^3\)  gridpoints are used for the velocity and temperature fields, and uniform grids with \(576^3\)  gridpoints for the phase field [1]}, which corresponds to at least 12 gridpoints used for the bubble height. The mesh is sufficiently fine and is comparable to corresponding single-phase studies [2]}, [3]}.
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